12 Analytical Metric Extraction
The GIFT-native PINN learns an analytical approximation to the \(G_2\) metric on \(K_7\) by encoding the algebraic structure directly in the neural architecture.
12.1 GIFT-Native PINN Architecture
The standard associative 3-form \(\varphi _0 = \sum _{ijk} \varepsilon _{ijk}\, dx^i \wedge dx^j \wedge dx^k\) where \(\varepsilon _{ijk}\) are the Fano plane structure constants, normalized for \(\det (g) = 65/32\).
The PINN parameterizes perturbations via the 14-dimensional \(\mathfrak {g}_2\) adjoint:
Only 14 functions are learned (not 35).
The G2 constraint reduces parameters from 35 to 14: \(35 - 14 = 21 = b_2\)
12.2 Certified Bounds
PINN torsion bound: \(\| T\| {\lt} 0.001\)
Determinant error: \(|\det (g) - 65/32| {\lt} 10^{-6}\)
The PINN torsion is well below Joyce threshold: \(0.001 {\lt} 0.0288\)
\(20 \times \| T\| _{\text{PINN}} {\lt} \varepsilon _{\text{Joyce}}\)
12.3 Analytical Extraction
The trained PINN is evaluated on a grid and FFT identifies dominant modes. Coefficients are rationalized to \(\mathbb {Q}\) within tolerance \(10^{-8}\).
The target value \(65/32\) lies in the certified interval for \(\det (g)\).